Scope

Construction of the real numbers

There are several ways of defining the real number system as an ordered field. The synthetic approach gives a list of axioms for the real numbers as a complete ordered field. Under the usual axioms of set theory, one can show that these axioms are categorical, in the sense that there is a model for the axioms, and any two such models are isomorphic. Any one of these models must be explicitly constructed, and most of these models are built using the basic properties of the rational number system as an ordered field. These constructions are described in more detail in the main article.

Continuity

A function from the set of real numbers to the real numbers can be represented by a graph in the Cartesian plane; such a function is continuous if, roughly speaking, the graph is a single unbroken curve with no "holes" or "jumps".

There are several ways to make this intuition mathematically rigorous. These definitions are equivalent to one another, so the most convenient definition can be used to determine whether a given function is continuous or not. In the definitions below,

is a function defined on a subsetI of the set R of real numbers. This subset I is referred to as the domain of f. Some possible choices include I=R, the whole set of real numbers, an open interval

Uniform continuity

If X and Y are subsets of the real numbers, a function f : X → Y is called uniformly continuous if for all ε > 0 there exists a δ > 0 such that for all x, y ∈ X, |x − y| < δ implies |f(x) − f(y)| < ε.

The difference between being uniformly continuous, and being simply continuous at every point, is that in uniform continuity the value of δ depends only on ε and not on the point in the domain.

Absolute continuity

Let be an interval in the real lineR. A function is absolutely continuous on if for every positive number , there is a positive number such that whenever a finite sequence of pairwise disjoint sub-intervals of satisfies

then

The collection of all absolutely continuous functions on I is denoted AC(I).

The following conditions on a real-valued function f on a compact interval [a,b] are equivalent:

(1) f is absolutely continuous;

(2) f has a derivative f ′ almost everywhere, the derivative is Lebesgue integrable, and

for all x on [a,b];

(3) there exists a Lebesgue integrable function g on [a,b] such that

for all x on [a,b].

If these equivalent conditions are satisfied then necessarily g = f ′ almost everywhere.

Equivalence between (1) and (3) is known as the fundamental theorem of Lebesgue integral calculus, due to Lebesgue.

Series

Given an infinitesequence of numbers { an }, a series is informally the result of adding all those terms together: a1 + a2 + a3 + · · ·. These can be written more compactly using the summation symbol ∑. An example is the famous series from Zeno's dichotomy and its mathematical representation:

The terms of the series are often produced according to a certain rule, such as by a formula, or by an algorithm.

Taylor series

where n! denotes the factorial of n and ƒ (n)(a) denotes the nth derivative of ƒ evaluated at the point a. The derivative of order zero ƒ is defined to be ƒ itself and (x − a)0 and 0! are both defined to be 1. In the case that a = 0, the series is also called a Maclaurin series.

Differentiation

Formally, the derivative of the function f at a is the limit

If the derivative exists everywhere, the function is differentiable. One can take higher derivatives as well, by iterating this process.

One can classify functions by their differentiability class. The class C0 consists of all continuous functions. The class C1 consists of all differentiable functions whose derivative is continuous; such functions are called continuously differentiable. Thus, a C1 function is exactly a function whose derivative exists and is of class C0. In general, the classes Ck can be defined recursively by declaring C0 to be the set of all continuous functions and declaring Ck for any positive integer k to be the set of all differentiable functions whose derivative is in Ck−1. In particular, Ck is contained in Ck−1 for every k, and there are examples to show that this containment is strict. C∞ is the intersection of the sets Ck as k varies over the non-negative integers. Cω is strictly contained in C∞.

Integration

Riemann integration

The Riemann integral is defined in terms of Riemann sums of functions with respect to tagged partitions of an interval. Let [a,b] be a closed interval of the real line; then a tagged partition of [a,b] is a finite sequence

This partitions the interval [a,b] into n sub-intervals [xi−1, xi] indexed by i, each of which is "tagged" with a distinguished point ti ∈ [xi−1, xi]. A Riemann sum of a function f with respect to such a tagged partition is defined as

thus each term of the sum is the area of a rectangle with height equal to the function value at the distinguished point of the given sub-interval, and width the same as the sub-interval width. Let Δi = xi−xi−1 be the width of sub-interval i; then the mesh of such a tagged partition is the width of the largest sub-interval formed by the partition, maxi=1...n Δi. The Riemann integral of a function f over the interval [a,b] is equal to S if:

For all ε > 0 there exists δ > 0 such that, for any tagged partition [a,b] with mesh less than δ, we have

When the chosen tags give the maximum (respectively, minimum) value of each interval, the Riemann sum becomes an upper (respectively, lower) Darboux sum, suggesting the close connection between the Riemann integral and the Darboux integral.

Lebesgue integration

Lebesgue integration is a mathematical construction that extends the integral to a larger class of functions; it also extends the domains on which these functions can be defined.

In real analysis, it is usually more natural to consider differentiable, smooth, or harmonic functions, which are more widely applicable, but may lack some more powerful properties of holomorphic functions. However, results such as the fundamental theorem of algebra are simpler when expressed in terms of complex numbers.

Techniques from the theory of analytic functions of a complex variable are often used in real analysis – such as evaluation of real integrals by residue calculus.